View source for 55edo

{{interwiki
| de = 55-EDO
| en = 55edo
| es = 55 EDO
| ja = 
}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
55edo can be used for a [[meantone]] tuning, and is close to [[1/6-comma meantone]] (and is almost exactly 10/57-comma meantone). {{w|Georg Philipp Telemann|Telemann}} suggested it as a theoretical basis for analyzing the [[meantone intervals|intervals of meantone]]. {{w|Leopold Mozart|Leopold}} and {{w|Wolfgang Amadeus Mozart|Wolfgang Mozart}} recommended 55edo or something close to it, with a subset and further approximation used for keyboard instruments which (apart from an experimental instrument) did not have enough notes per octave to accommodate it in full.<ref>Chesnut, John (1977) ''Mozart's Teaching of Intonation'', '''Journal of the American Musicological Society''' Vol. 30, No. 2 (Summer, 1977), pp. 254-271 (Published By: University of California Press) [https://doi.org/10.2307/831219 doi.org/10.2307/831219], [http://www.jstor.org/stable/831219 https://www.jstor.org/stable/831219]</ref> It can also be used for [[Meantone_family#Mohajira|Mohajira]] and [[Meantone_family#Liese|Liese]] temperaments. It also supports an extremely sharp tuning of [[huygens|Huygens/undecimal meantone]] using the 55de [[val]], meaning that primes 7 and 11 are mapped very sharply to their second-best mapping.

=== Odd harmonics ===
{{Harmonics in equal|55}}

=== Subsets and supersets ===
Since 55 factors into {{factorization|55}}, 55edo contains [[5edo]] and [[11edo]] as its subsets.

== Intervals ==
{| class="wikitable center-1 right-2 left-3"
|-
! [[Degree|&#35;]]
! [[Cent]]s
! Approximate ratios
! colspan="3" | [[Ups and downs notation]]
|-
| 0
| 0.0
| 1/1
| P1
| perfect 1sn
| D
|-
| 1
| 21.8
| 65/64, 78/77, 99/98, ''128/125''
| ^1
| up 1sn
| ^D
|-
| 2
| 43.6
| 36/35, ''64/63''
| ^^1
| dup 1sn
| ^^D
|-
| 3
| 65.5
| 28/27
| vvm2
| dudminor 2nd
| vvEb
|-
| 4
| 87.3
| 21/20, ''18/17'', ''25/24''
| vm2
| downminor 2nd
| vEb
|-
| 5
| 109.1
| 16/15, 17/16
| m2
| minor 2nd
| Eb
|-
| 6
| 130.9
| 13/12, 14/13
| ^m2
| upminor 2nd
| ^Eb
|-
| 7
| 152.7
| 12/11, ''11/10''
| ~2
| mid 2nd
| vvE
|-
| 8
| 174.5
| 
| vM2
| downmajor 2nd
| vE
|-
| 9
| 196.4
| 9/8, ''10/9''
| M2
| major 2nd
| E
|-
| 10
| 218.2
| 17/15
| ^M2
| upmajor 2nd
| ^E
|-
| 11
| 240.0
| 8/7
| ^^M2
| dupmajor 2nd
| ^^E
|-
| 12
| 261.8
| 7/6
| vvm3
| dudminor 3rd
| vvF
|-
| 13
| 283.6
| 13/11
| vm3
| downminor 3rd
| vF
|-
| 14
| 305.5
| 6/5
| m3
| minor 3rd
| F
|-
| 15
| 327.3
| 
| ^m3
| upminor 3rd
| ^F
|-
| 16
| 349.1
| 11/9, 27/22
| ~3
| mid 3rd
| ^^F
|-
| 17
| 370.9
| 26/21, ''16/13''
| vM3
| downmajor 3rd
| vF#
|-
| 18
| 392.7
| 5/4
| M3
| major 3rd
| F#
|-
| 19
| 414.5
| 14/11
| ^M3
| upmajor 3rd
| ^F#
|-
| 20
| 436.4
| 9/7
| ^^M3
| dupmajor 3rd
| ^^F#
|-
| 21
| 458.2
| ''21/16''
| vv4
| dud 4th
| vvG
|-
| 22
| 480.0
| 
| v4
| down 4th
| vG
|-
| 23
| 501.8
| 4/3, ''27/20''
| P4
| perfect 4th
| G
|-
| 24
| 523.6
| 
| ^4
| up 4th
| ^G
|-
| 25
| 545.5
| 11/8, 15/11
| ~4
| mid 4th
| ^^G
|-
| 26
| 567.3
| [[7/5]], [[18/13]]
| vA4
| downaug 4th
| vG#
|-
| 27
| 589.1
| 24/17
| A4, vd5
| aug 4th, downdim 5th
| G#, vAb
|-
| 28
| 610.9
| 17/12
| ^A4, d5
| upaug 4th, dim 5th
| ^G#, Ab
|-
| 29
| 632.7
| [[10/7]], [[13/9]]
| ^d5
| updim 5th
| ^Ab
|-
| 30
| 654.5
| 16/11, 22/15
| ~5
| mid 5th
| vvA
|-
| 31
| 676.4
| 
| v5
| down 5th
| vA
|-
| 32
| 698.2
| 3/2, ''40/27''
| P5
| perfect 5th
| A
|-
| 33
| 720.0
| 
| ^5
| up 5th
| ^A
|-
| 34
| 741.8
| ''32/21''
| ^^5
| dup 5th
| ^^A
|-
| 35
| 763.6
| 14/9
| vvm6
| dudminor 6th
| vvBb
|-
| 36
| 785.5
| 11/7
| vm6
| downminor 6th
| vBb
|-
| 37
| 807.3
| 8/5
| m6
| minor 6th
| Bb
|-
| 38
| 829.1
| 21/13, ''13/8''
| ^m6
| upminor 6th
| ^Bb
|-
| 39
| 850.9
| 18/11, 44/27
| ~6
| mid 6th
| vvB
|-
| 40
| 872.7
| 
| vM6
| downmajor 6th
| vB
|-
| 41
| 894.5
| 5/3
| M6
| major 6th
| B
|-
| 42
| 916.4
| 22/13
| ^M6
| upmajor 6th
| ^B
|-
| 43
| 938.2
| 12/7
| ^^M6
| dupmajor 6th
| ^^B
|-
| 44
| 960.0
| 7/4
| vvm7
| dudminor 7th
| vvC
|-
| 45
| 981.8
| 30/17
| vm7
| downminor 7th
| vC
|-
| 46
| 1003.6
| 16/9, ''9/5''
| m7
| minor 7th
| C
|-
| 47
| 1025.5
| 
| ^m7
| upminor 7th
| ^C
|-
| 48
| 1047.3
| 11/6, ''20/11''
| ~7
| mid 7th
| ^^C
|-
| 49
| 1069.1
| 13/7, 24/13
| vM7
| downmajor 7th
| vC#
|-
| 50
| 1090.9
| 15/8, ''32/17''
| M7
| major 7th
| C#
|-
| 51
| 1112.7
| 40/21, ''17/9'', ''48/25''
| ^M7
| upmajor 7th
| ^C#
|-
| 52
| 1134.5
| 56/27
| ^^M7
| dupmajor 7th
| ^^C#
|-
| 53
| 1156.4
| 35/18, ''63/32''
| vv8
| dud 8ve
| vvD
|-
| 54
| 1178.2
| 128/65, 77/39, 196/99, ''125/64''
| v8
| down 8ve
| vD
|-
| 55
| 1200.0
| 2/1
| P8
| perfect 8ve
| D
|}
<nowiki />* 55f val (tending flat), inconsistent intervals labeled in ''italic''

== Notation ==
=== Ups and downs notation ===
55edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that dup is equivalent to dudsharp and dud is equivalent to dupflat.
{{Sharpness-sharp4a}}
[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
{{Sharpness-sharp4}}
=== Sagittal notation ===
==== Evo flavor ====
<imagemap>
File:55-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 615 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:55-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Revo_Sagittal.svg]]
</imagemap>

==== Evo-SZ flavor ====
<imagemap>
File:55-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[33/32]]
default [[File:55-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

=== 31-tone subset ===
The 31-out-of-55edo subset can be notated entirely with the standard notation of 7 each of naturals/sharps/flats, and 5 each of doublesharps/doubleflats, as a 31-tone chain-of-5ths from Gbb to Ax.

[[File:Monzo55Notation.jpeg|400px|frameless|alt=Diagram of 31-tone subset of 55edo using plain Western notation, by Joe Monzo.|Diagram of 31-tone subset of 55edo using plain Western notation, by [[Joe Monzo]].]]

== Approximation to JI ==
[[File:55ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 55edo]]

=== Selected just intervals by error ===
{{Q-odd-limit intervals|55}}
{{Q-odd-limit intervals|55.05|apx=val|header=none|tag=none|title=15-odd-limit intervals by 55d val mapping}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -87 55 }}
| {{mapping| 55 87 }}
| +1.31
| 1.19
| 7.21
|-
| 2.3.5
| 81/80, {{monzo| 31 1 -14 }}
| {{mapping| 55 87 128 }}
| −0.13
| 2.10
| 9.63
|}

=== Uniform maps ===
{{Uniform map|edo=55}}

=== Commas ===
{{Todo|cleanup|inline=true}}

'''5-limit commas''': [[81/80]], [[Quintosec_family|{{monzo| 47 -15 -10 }}]], {{monzo| 31 1 -14 }}, {{monzo| 27 5 -15 }}

'''7-limit commas''': 31104/30625, [[6144/6125]], 81648/78125, 16128/15625, 28672/28125, 33075/32768, 83349/80000, 1029/1000, [[686/675]], [[10976/10935]], [[Cloudy comma|16807/16384]], 84035/82944

'''11-limit commas''': 59049/58564, 74088/73205, 46656/46585, 21609/21296, 12005/11979, 19683/19360, [[243/242]], 3087/3025, 5488/5445, 19683/19250, 1944/1925, 45927/45056, 2835/2816, 35721/34375, 7056/6875, 12544/12375, 7203/7040, 2401/2376, 24057/24010, 72171/70000, 891/875, [[176/175]], 2079/2048, [[385/384]], 3234/3125, 17248/16875, 26411/25600, 26411/2592, 26411/262404, 88209/87808, 30976/30625, 3267/3200, [[121/120]], 81312/78125, 41503/40000, 41503/40500, 35937/35000, 2662/2625, 42592/42525, 83853/81920, 9317/9216, 65219/62500, 43923/43904, 14641/14400, [[14641/14580]]

'''13-limit commas''': 59535/57122, 29400/28561, 29568/28561, 29645/28561, 24576/24167, 99225/96668, 24500/24167, 50421/48334, 45927/43940, 2268/2197, 2240/2197, 57624/54925, 61875/61516, 57024/54925, 11264/10985, 72765/70304, 13475/13182, 22869/21970, 6776/6591, 20736/20449, 20480/20449, 84035/81796, 91125/91091, 65536/65065, 15309/14872, 1890/1859, 5600/5577, 9604/9295, 59049/57967, 58320/57967, 4374/4225, 864/845, [[512/507]], 11025/10816, 6125/6084, 21952/21125, 16807/16224, 84035/82134, 66825/66248, 90112/88725, 56133/54080, 693/676, 1540/1521, 26411/25350, 58806/57967, 58080/57967, 88209/84500, 4356/4225, 7744/7605, 88935/86528, 33275/33124, 27951/27040, 9317/9126, 58564/57967, 43923/42250, 17496/17303, 87808/86515, 55296/55055, 25515/25168, [[1575/1573]], 64827/62920, 4802/4719, 98415/98098, 59049/57200, 729/715, [[144/143]], 18375/18304, 18522/17875, 10976/10725, 84035/82368, 59049/56875, 11664/11375, 2304/2275, [[4096/4095]], 1701/1664, [[105/104]], 42336/40625, 25088/24375, 21609/20800, 2401/2340, 9604/9477, 72171/71344, 2673/2600, [[66/65]], [[352/351]], 13475/13312, 33957/32500, 15092/14625, 81675/81536, 58806/56875, 11616/11375, 61952/61425, 68607/66560, 847/832, 4235/4212, 35937/35672, 1331/1300, 5324/5265, 58564/56875, 85293/85184, 13377/13310, 85293/84700, 15288/15125, 31213/30976, 67392/67375, 28431/28160, 34944/34375, 4459/4400, 4459/4455, 28431/28000, [[351/350]], 79872/78125, 66339/65536, 51597/50000, 637/625, 10192/10125, 31213/30720, [[31213/31104]], 30888/30625, 1287/1280, 81081/78125, 16016/15625, 49049/48000, 49049/48600, 14157/14000, 33033/32768, 77077/75000, 51909/51200, 17303/17280, 75712/75625, 8281/8250, 41067/40960, 31941/31250, 9464/9375, 57967/57600, 91091/90000, 61347/61250, 79092/78125

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 6\55
| 130.9
| 14/13
| [[Twothirdtonic]] (55f)
|-
|1
|8\55
|174.5
|[[10/9]]~[[11/10]]
|[[Tetracot]] (55c)
|-
| 1
| 16\55
| 349.1
| 11/9
| [[Mohaha]]
|-
| 1
| 23\55
| 501.8
| 4/3
| [[Meantone]] (55d)
|-
| 1
| 26\55
| 567.3
| 7/5
| [[Liese]] (55)
|-
| 1
| 27\55
| 589.1
| 45/32
| [[Untriton]] (55d) / [[aufo]] (55)
|-
| 5
| 17\55<br>(5\55)
| 370.9<br>(109.1)
| 99/80<br>(16/15)
| [[Quintosec]]
|-
| 11
| 23\55<br>(3\55)
| 501.8<br>(65.5)
| 4/3<br>(36/35)
| [[Hendecatonic]] (55)
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==
; Subsets of twothirdtonic[37]
* Undecimal otonal-like pentatonic: 17 8 7 12 11

; Subsets of hendecatonic[33]
* Septimal pentatonic-like: 10 13 9 13 10
* Septimal minor blues-like: 13 10 4 5 13 10
* Septimal heptatonic blues-like: 13 10 4 5 8 5 10

; Others
* Sakura-like scale containing [[phi]]: 9 6 18 5 17
* Quasi-[[equiheptatonic]] scale: 8 8 7 9 7 9 7

== Instruments ==
* [[Lumatone mapping for 55edo]]

== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=oymJKnYzzOw "Jesus bleibet meine Freude" from ''Herz und Mund und Tat und Leben'', BWV 147] (1723) – arranged for two organs, rendered by Claudi Meneghin (2021)
* [https://www.youtube.com/watch?v=xoCNOIsjfeU "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by [[Claudi Meneghin]] (2024)
* [https://www.youtube.com/watch?v=OkRVNo19guo "Ricercar a 6" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2025)
* [https://www.youtube.com/watch?v=Y5sIjh_Te40 "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=QOPxqNgkVWM "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)

; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=OfOt3nOp-f8 ''Prelude in E Minor "The Great"''] – rendered by [[Claudi Meneghin]] (2023)
* [https://www.youtube.com/watch?v=tuIPIhSxUPs ''Prelude in E Minor "The Little"''] – rendered by Claudi Meneghin (2024)

; {{W|Georg Frideric Handel}}
* [https://www.youtube.com/watch?v=rDvKPuzsno8 ''Fugue'' from "Suite in E minor", HWV 429] (1720) – arranged for Baroque ensemble and drums, rendered by Claudi Meneghin (2025) 

; {{W|Scott Joplin}}
* [https://www.youtube.com/watch?v=GbhpuoIJgxk ''Maple Leaf Rag''] (1899) – arranged for harpsichord and rendered by [[Claudi Meneghin]] (2024)

; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=C_AML6XW-2g ''Rondo alla Turca'' from the Piano Sonata No. 11, KV 331] (1778) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=XgRksdk6zyQ ''Fugue in G minor'', KV 401] (1782) – rendered by Francium (2023)
* [http://www.seraph.it/dep/int/AdagioKV540.mp3 ''Adagio in B minor'', KV 540] (1788) – rendered by Carlo Serafini (2011) ([http://www.seraph.it/blog_files/706c4662272db7703def4d57edfcb955-119.html blog entry])
* [https://www.youtube.com/watch?v=pFjJCj2MBTM ''Allegro'' from the Piano Sonata No. 16, KV 545] (1788) – rendered by Francium (2023)
* [https://www.youtube.com/watch?v=p88MWgdio14&list=PLC6ZSKWKnVz0mOTLQkCUi9ydWGLpBP8gZ&index=2 ''Mozart's Gigue KV 574, Arranged for Fortepiano (55-edo)''] – rendered by [[Claudi Meneghin]] (2025)

; {{W|Keiichi Okabe}}
* [https://www.youtube.com/watch?v=L24G4Y7tZgI ''Yuutsu no Yuutsu''] (2006) – rendered by MortisTheneRd (2024)

=== 21st century ===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/l62rb8ULCXs ''55edo improv''] (2025)
* [https://www.youtube.com/watch?v=kVmToKkZU88 ''Waltz in 55edo''] (2025)

; [[James Kukula]]
* ''[https://app.box.com/s/8hq89cb3rqqkrhvkxgvqtppa255kcqrq?fbclid=IwY2xjawISjSlleHRuA2FlbQIxMAABHcl5t8n_C7QUJqdEnwSaWBc5u3BpldmcAjhQQljsQIPl1qJ-zdCr9T8NMw_aem_Ez0m-Ls_ZqI0-c0Ld-28Yg 55edo Melted Syntonic]'' (2025)

; [[Budjarn Lambeth]]
* ''[https://www.youtube.com/watch?v=9c5MtrZFNhA Improvisation One in 55edo]'' (2025)
* ''[https://www.youtube.com/watch?v=ggFGUn1Ya2A Improvisation Two in 55edo]'' (2025)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=AgsJCTyxqiM ''Double Fugue on "We Wish You a Merry Christmas" for String Quartet''] (2020)
* [https://www.youtube.com/watch?v=rAbbvyotIr4 ''Canon at the Diatonic Semitone on an Ancient Lombard Theme''] (2021)
* [https://www.youtube.com/watch?v=hCUIx1RzvEk ''Chacony "Lament & Deception"'' for Two Violins and Cello] (2021), [https://www.youtube.com/watch?v=abJP4euMlsg for Baroque Wind Ensemble] (2023)
* [https://www.youtube.com/watch?v=9zfWeO0eJdA Fantasy "Almost a Fugue" on a Theme by Giuliani, for String Quartet] (2021)
* [https://www.youtube.com/watch?v=jOiub14Cskw ''Double Fugue on "Old McDonald" + "Shave & a Haircut"''] (2024)

; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/road-trip-to-nowhere Road Trip to Nowhere]'' (2021)
* ''[https://soundcloud.com/morphosyntax-1/migration Migration]'' (2025)

== External links ==
* ''[http://tonalsoft.com/monzo/55edo/55edo.aspx Mozart's tuning: 55-edo and its close relative, 1/6-comma meantone]'' (containing another listening example) on [[Tonalsoft Encyclopedia]]

== References ==
<references />

[[Category:Meantone]]
[[Category:Historical]]
[[Category:Listen]]